How would I twist a torus or circle into an infinity symbol?
One way would be to do something similar to this, but with fewer twists.

I found that  100&deg; twist with the simple deform modifier gives a pretty good infinity symbol when seen from the top:



By animating the amount of twisting (you can hover over almost any UI element and press I to insert a keyframe), you can get a result like this:



Example .blend